A universal complementarity identity for polarized double-slit interferometry

Abstract

An exact identity is established among four experimentally accessible quantities in polarized double-slit interferometry: the phase-reference-dependent in-phase and quadrature components $V_A$ and $V_N$ of fringe visibility, the path predictability $\mathcal{P}$, and the mixedness $\mathcal{I}$ of the reduced path state satisfy $V_A^2+V_N^2+\mathcal{P}^2+\mathcal{I}^2=1$. The identity is an algebraic consequence of positivity and holds for every normalized path--polarization density matrix. It contains the Greenberger--Yasin predictability bound and, for globally pure path--polarization states, the Jakob--Bergou complete-complementarity equality; it is also connected with Englert's distinguishability relation when polarization carries which-path information. The separation $V^2=V_A^2+V_N^2$ resolves visibility into two components measurable by phase-shifted interferometry. Within a fixed real basis and a fixed phase convention, the quadrature-sensitive component is read from the antisymmetric sector of the Hermitian decomposition $ρ=A+iN$. A maximum-entropy reconstruction is included as an interpretation of how measurements sensitive to the two sectors constrain an inferred state, but the identity itself does not depend on that reconstruction.